Principal component analysis for interval data

Abstract

Principal component analysis for classical data is a method used frequently to reduce the effective dimension underlying a data set from p random variables to s p linear functions of those p random variables and their observed values. With contemporary large data sets, it is often the case that the data are aggregated in some meaningful scientific way such that the resulting data are symbolic data (such as lists, intervals, histograms, and the like); though symbolic data can and do occur naturally and in smaller data sets. Since symbolic data have internal variations along with the familiar (between observations) variation of classical data, direct application of classical methods to symbolic data will ignore much of the information contained in the data. Our focus is to describe and illustrate principal component methodology for interval data. The significance of symbolic data in general and of this article in particular is illustrated by its applicability for our analysis of three key 21st century challengers: networks, security data, and translational medicine. It is relatively easy to visualize the applicability to security data and translational medicine, though less easy to visualize its applicability to networks. Since an interval is typically denoted by (a, b), in a network interval, we let a be a pair of nodes and b be their edge with characteristics c and d, respectively. If this representation of a network interval is valid, then we can more easily visualize its applicability to networks also.

Matrix Factorization with Interval-Valued Data

“…Consequently, several data analysis tools, including regression [23], [24], canonical analysis [25], and multi-dimensional scaling [26], have been developed for symbolic and interval-valued data. Given the popularity of PCA in data analysis, several interval-valued PCA algorithms have also been proposed [27], [28], [29], [30], most of which leverage the specific statistical and geometric meanings of principal components of a system of variables. As discussed above, interval NMF and PMF [9] also have been studied to resolve alignment approximation in face analysis and rating approximation in collaborative filtering.…”

Fault Detection of Single and Interval Valued Data Using Statistical Process Monitoring Techniques

“…Initially, the use of interval data is motivated by the need to quickly and efficiently monitor large datasets [28], in addition to its ability to deal with missing values without the need to remove entire samples. Generating intervals by aggregation is a form of batch processing, which may not always be ideal.…”

Matrix Factorization with Interval-Valued Data

“…Consequently, several data analysis tools, including regression [23], [24], canonical analysis [25], and multi-dimensional scaling [26], have been developed for symbolic and interval-valued data. Given the popularity of PCA in data analysis, several interval-valued PCA algorithms have also been proposed [27], [28], [29], [30], most of which leverage the specific statistical and geometric meanings of principal components of a system of variables. As discussed above, interval NMF and PMF [9] also have In contrast, we develop a more general interval-valued latent semantic alignment algorithm which can be integrated in common matrix factorization approaches that directly leverages interval-valued properties.…”